Abstract Algebra 13.4: A Polynomial Factor Ring

In this video, we consider a more complicated example of a factor ring, and show how it is effectively the same as the complex numbers.

[n.trushaev]

Great video. I have been getting legitimately frustrated and angry because I'm doing online classes now due to the ongoing corona lockdown, and my prof. did not thoroughly cover the details regarding the structure of polynomial factor rings or provide any lecture material, but basically told us to use our textbook to work out the computational details. It's very frustrating because textbooks usually don't go very deep into the details of their computations or examples, and there are very few video resources available for abstract algebra and other mid-to-high level mathematics.

[benjaminmartin]

Thank you great video explaining how quotients in polynomial rings actually works.

[silesiaball]

I have a question, because I'm not sure if I understand this.
Why can we say that x^2+1 is effectively 0?

[algebraentodaspartes]

Thanks a lot, I'm starting to study rings on my own and this helped me a lot with the examples in my book.

[ibrahimislam]

Dear Patrick, ....I have a question from this topic could you please help me, ....my question is: what is ring ℤₚ[x]) / ⟨x²⟩, , ...I mean to what ring it looks like which is easy to see, ....is it isomorphic to ℤₚ² ?I have strong intuttion that it is isomorphic to ℤₚ² but can't prove it is not isomorphic to ℤₚ², then to which ring it is isomorphic?

[athair]

Hi; i take this video as my main reference on the subject, so well explained that i keep coming back from time to time; i would like to ask if it could be right to say there is a hint on the absortion property of ideals in it? i mean an absortion property such that operations on the quotient turn back to the ideal as the zero-class(?), in the sense of an additive (group) zero class (?) thanks a lot once again!!

[hussainibnali]

Thank you soo much for cleaning my dought's 😊 I need this

[darpi_]

I had been stuck here for a while. Thank you so much for clearing my doubt! I have my algebra exam tomorrow wish me luck !

[xs]

good, now i understand why it can be treat as complex numbers

[OpelsiSq]

I was blown away when i realized about it being isomorphic to complex numbers. Somehow i didn't notice x^2=-1 (which means x=i) until you metioned it)

[athair]

A rather dumb question probably: what are some applications of a polinomyal quotient ring (beside the homomorphism' theorem.that states that a (given) ring homomorphism is isomorphic to the quotient ring defined by the ideal? Thanks